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Gröbner-Shirshov bases and their calculation. (English) Zbl 1350.13001

The article under review is an excellent survey on the theory of Gröbner-Shirshov bases for different classes of linear universal algebras, together with an overview of the calculation of these bases in a variety of specific cases. The main problem discussed is the following fundamental question: How to find a linear basis of an algebra defined by generators and relations?
For commutative associative algebras presented as factor algebras of polynomial algebras the solution is given by the famous Buchberger algorithm from 1965. Starting with an arbitrary generating set of an ideal the algorithm gives the Gröbner basis of the ideal. Then the basis of the factor algebra consists of all monomials which do not contain as a subword a leading monomial of a polynomial from the Gröbner basis. A similar idea was developed by Shirshov in 1962 for Lie algebras. It is clear from his exposition that the same methods work also for (noncommutative) associative algebras.
Later Bergman in 1978 used the Diamond Lemma from graph theory to solve the problem how to find a basis of an associative algebra defined by generators and relations.
Before its English translation [SIGSAM Bull. 33, No. 2, 3–6 (1999; Zbl 1097.17502)], the paper [Sib. Mat. Zh. 3, 292–296 (1962; Zbl 0104.26004)] was practically unknown outside East Europe. Honoring the achievements of Shirshov, the first author of the survey suggested the name Gröbner-Shirshov bases for the noncommutative and nonassociative analogues of the Gröbner bases in commutative algebra. In the introduction the authors present a detailed historical overview on the developing of the methods to solve the main problem under discussion and give the range of the applications of versions of the Gröbner-Shirshov bases for different classes of algebras. The second section is devoted to associative algebras, the third section deals with semigroups and groups, the forth section considers Lie algebras, and the final section studies \(\Omega\)-algebras and operads.
The article is completed with a long list of references containing 228 titles.

MSC:

13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
17B01 Identities, free Lie (super)algebras
17A30 Nonassociative algebras satisfying other identities
18D50 Operads (MSC2010)
20M18 Inverse semigroups
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Full Text: DOI arXiv

References:

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